Mathematics 133 -- Intensive Calculus for Science 1

Lab 1: Getting Started with Maple, Plotting Functions and Derivatives

October 12, 2001

Goals

Today, we will begin using a program called Maple to draw graphs of functions of one variable (given by symbolic formulas) and their derivatives. You will be using the commands covered here for most of the other lab days this semester. On the mathematical side, we will study the relation between functions and their derivative functions.

Maple Background and A First Example

Examples of all of the commands described here may be found in the handouts from yesterday's class. You will probably want to consult them for additional information.

First, you will need to get into Windows and Maple as described in the General Information handout. The basic Maple command for 2D plotting graphs of the form y = f(x) is called plot. The basic format is

plot(function,range,options);

where

  1. function is the function to be plotted. You will always want to define this first using the function definition format given in the handout of Maple examples.
  2. range is the range of x-values you want to see plotted, and
  3. options can be used to control the form of the plot if desired. No options need be specified however if you don't want to. More on this later.

The above plot command is a template. To do an actual plot, you will need to fill in the function and range of x values for the plot you want and add the appropriate options. For example, suppose we wanted to plot y = x4 - 2x3 + x - 5 for x between -2 and 1. We could use the Maple plot command with no options:

f:=x->x^4-2*x^3+x-5;

plot(f(x),x=-2..1);

(Note: The close parenthesis comes directly after the range of x-values if there are no options.) Type in these command lines (exactly as here) and press ENTER (RETURN) after each one. If you make a typing mistake, Maple will let you know about it(!) Fortunately, if this happens, the whole command does NOT need to be re-entered. Just move the cursor arrow to the place on the input line you want to change, press the left mouse button, and edit the input as needed, then press RETURN again

When all goes well you will see the graphics output displayed under the input command in the worksheet.

You can display several plots together by putting the formulas for the functions together inside the plot command, in a set of curly brackets ( { } ), separated by commas. For example, try entering the command

plot({f(x),cos(x)},x=-2..1);

to plot our polynomial function from before together with y = cos(x). The cosine function is built-in in Maple.

Let's add one more thing. Sometimes, an informative title makes a graph much more understandable. To add a text title to a plot, you can insert a comma after the range of x-values, and include an option in the plot command of the form

title="whatever you want"

Some additional information:
  1. Maple ``knows'' all the functions we studied in Chapter 1 of our text. The names of the most common ones are sin, cos, tan, exp, ln. To use one of these functions in a Maple formula, you put the name, followed by the "argument" (that is the expression you are applying the function to) in parentheses.
  2. The range of x-values to plot for a graph y = f(x) must be specified, in the format x = low..high. A range of y-values to plot can also be specified, in the same format.
  3. Every Maple command must be terminated either with a semicolon or a colon. This is the signal that the command is finished and that Maple should try to execute what you asked it to do. The difference between a semicolon and a colon is that for commands terminated with a semicolon, any output produced will be displayed. If you terminate the command with a colon, the calculation will be performed but not displayed (this is useful sometimes for intermediate steps in a big computation where you don't need to see the output).

Recall that we have introduced the derivative function f'(x) for a function f(x). In Maple, the derivative function of f is D(f).

Lab Questions

In these questions, we will study the function

f(x) = sin(x) - x cos(2 x).

and its derivative, etc. To plot this function in Maple, you will need to translate the mathematical formula into a Maple expression, and define a function as in the handout of examples. Do this first.

  1. Plot the portion of the graph y = f(x) on the x-range -4 <= x <= 4.
  2. Now, plot the portion of the graph y = f'(x) for -4 <= x <= 4, on a separate set of axes.
  3. Use the graph of y = f'(x) to estimate the value of f'(1). Use that value to find the equation of the tangent line to y=f(x) at (1,f(1)) (do this by hand), and plot that tangent line together with y = f(x).
  4. Answer the following questions in a text region in the worksheet -- see the Maple Information handout for how to create these:
  5. The derivative of the derivative of a function f(x) is called the second derivative of f, written f''(x). In Maple, the second derivative function of f is (D@@2)(f) (note the extra set of parentheses around the D@@2 -- they must be included). Plot the second derivative of f(x) = sin(x) - x cos(2 x) on the same interval as in the previous questions.
  6. Answer the following questions in a second text region in your worksheet:

Assignment

The lab write-up (a paper print-out of your worksheet, including all graphs and explanations in text regions) is due on Wednesday, October 17. Do not include the worked example, and resize all of your plots smaller than the default output plot size that Maple gives you to conserve paper when you print.